Heat oil in a large heavy pot over medium heat. Add curry paste and cook, stirring, until fragrant, about 1 minute. Add carrots, onion, and pepper and cook, stirring occasionally, until onion is translucent, about 10 minutes.
Add potatoes, chicken, coconut milk, and 1 1/2 cups water and bring to a boil. Reduce heat to a simmer and cook, stirring occasionally, until chicken is cooked through and potatoes are tender, about 20 minutes. Divide curry among bowls and top with herbs.
